#### Definition 5.3

*Let* (*X*, *T*, *F*) *be a ternary f*-*rack*, *A be an abelian group and f*, *g*: *X* → *X be homomorphisms*. *A structure of X*-*module on A consists of a family of automorphisms* (*η*_{ijk})_{i, j, k ∈ X} *and a family of endomorphisms* (*τ*_{ijk})_{i, j, k ∈ X} *of A satisfying the following conditions*:
$$\begin{array}{}{\eta}_{T(x,y,z),f(u),f(v)}{\eta}_{x,y,z}={\eta}_{T(x,u,v),T(y,u,v),T(z,u,v)}{\eta}_{x,u,v}\end{array}$$(8)
$$\begin{array}{}{\eta}_{T(x,y,z),f(u),f(v)}{\tau}_{x,y,z}={\tau}_{T(x,u,v),T(y,u,v),T(z,u,v)}{\eta}_{y,u,v}\end{array}$$(9)
$$\begin{array}{}{\eta}_{T(x,y,z),f(u),f(v)}{\mu}_{x,y,z}={\mu}_{T(x,u,v),T(y,u,v),T(z,u,v)}{\eta}_{z,u,v}\end{array}$$(10)
$$\begin{array}{}{\tau}_{T(x,y,z),f(u),f(v)}g={\eta}_{T(x,u,v),T(y,u,v),T(z,u,v)}{\tau}_{x,u,v}+{\tau}_{T(x,u,v),T(y,u,v),T(z,u,v)}{\tau}_{y,u,v}\\ +{\mu}_{T(x,u,v),T(y,u,v),T(z,u,v)}{\tau}_{z,u,v}\end{array}$$(11)
$$\begin{array}{}{\mu}_{T(x,y,z),f(u),f(v)}g={\eta}_{T(x,u,v),T(y,u,v),T(z,u,v)}{\mu}_{x,u,v}+{\tau}_{T(x,u,v),T(y,u,v),T(z,u,v)}{\mu}_{y,u,v}\\ +{\mu}_{T(x,u,v),T(y,u,v),T(z,u,v)}{\mu}_{z,u,v}\end{array}$$(12)

In the *n*-ary case, we generalized the above definition as follows.

#### Definition 5.4

*Let* (*X*, *T*, *F*) *be an n*-*ary f*-*rack*, *A be an abelian group and f*, *g*: *X* → *X be homomorphisms*. *A structure of X*-*module on A consists of a family of automorphisms* (*η*_{ijk})_{i, j, k ∈ X} *and a family of endomorphisms*
$\begin{array}{}({\tau}_{ijk}^{i}{)}_{i,j,k\in X}\end{array}$ *of A satisfying the following conditions*:
$$\begin{array}{}{\eta}_{T({x}_{1},{x}_{2},...,{x}_{n}),f({y}_{2}),f({y}_{3}),...,f({y}_{n})}{\eta}_{{x}_{1},{x}_{2},...,{x}_{n}}={\eta}_{T({x}_{1},y2,...,{y}_{n}),T({x}_{2},{y}_{2},...,{y}_{n}),...,T({x}_{n},{y}_{2},...,{y}_{n})}{\eta}_{{x}_{1},{y}_{2},...,{y}_{n}}\end{array}$$(13)
$$\begin{array}{}{\eta}_{T({x}_{1},{x}_{2},...,{x}_{n}),f({y}_{2}),f({y}_{3}),...,f({y}_{n})}{\tau}_{{x}_{1},{x}_{2},...,{x}_{n}}^{i}={\tau}_{T({x}_{1},y2,...,{y}_{n}),T({x}_{2},{y}_{2},...,{y}_{n}),...,T({x}_{n},{y}_{2},...,{y}_{n})}^{i}{\eta}_{{x}_{i},{y}_{2},...,{y}_{n}}\end{array}$$(14)
$$\begin{array}{}{\tau}_{T({x}_{1},{x}_{2},...,{x}_{n}),f({y}_{2}),f({y}_{3}),...,f({y}_{n})}^{i}g={\eta}_{T({x}_{1},y2,...,{y}_{n}),T({x}_{2},{y}_{2},...,{y}_{n}),...,T({x}_{n},{y}_{2},...,{y}_{n})}{\tau}_{{x}_{1},{y}_{2},...,{y}_{n}}^{i}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}+\sum _{j=1}^{n-1}{\tau}_{T({x}_{1},y2,...,{y}_{n}),T({x}_{2},{y}_{2},...,{y}_{n}),...,T({x}_{n},{y}_{2},...,{y}_{n})}^{j}{\tau}_{{x}_{j},{y}_{2},...,{y}_{n}}^{i}.}\end{array}$$(15)

#### Example 5.7

*Let A be a non-empty set and* (*X*, *T*, *F*) *be a ternary f*-*quandle*, *and κ be a generalized 2-cocycle*. *For a*, *b*, *c* ∈ *A*, *let*
$$\begin{array}{}{\alpha}_{x,y,z}(a,b,c)={\eta}_{x,y,z}(a)+{\tau}_{x,y,z}(b)+{\mu}_{x,y,z}(c)+{\kappa}_{x,y,z}.\end{array}$$

*Then*, *it can be verified directly that α is a dynamical cocycle and the following relations hold*:
$$\begin{array}{}{\eta}_{T(x,y,z),f(u),f(v)}{\eta}_{x,y,z}={\eta}_{T(x,u,v),T(y,u,v),T(z,u,v)}{\eta}_{x,u,v}\end{array}$$(18)
$$\begin{array}{}{\eta}_{T(x,y,z),f(u),f(v)}{\tau}_{x,y,z}={\tau}_{T(x,u,v),T(y,u,v),T(z,u,v)}{\eta}_{y,u,v}\end{array}$$(19)
$$\begin{array}{}{\eta}_{T(x,y,z),f(u),f(v)}{\mu}_{x,y,z}={\mu}_{T(x,u,v),T(y,u,v),T(z,u,v)}{\eta}_{z,u,v}\end{array}$$(20)
$$\begin{array}{}{\tau}_{T(x,y,z),f(u),f(v)}g={\eta}_{T(x,u,v),T(y,u,v),T(z,u,v)}{\tau}_{x,u,v}+{\tau}_{T(x,u,v),T(y,u,v),T(z,u,v)}{\tau}_{y,u,v}\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}+{\mu}_{T(x,u,v),T(y,u,v),T(z,u,v)}{\tau}_{z,u,v}\end{array}$$(21)
$$\begin{array}{}{\mu}_{T(x,y,z),f(u),f(v)}g={\eta}_{T(x,u,v),T(y,u,v),T(z,u,v)}{\mu}_{x,u,v}+{\tau}_{T(x,u,v),T(y,u,v),T(z,u,v)}{\mu}_{y,u,v}\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}+{\mu}_{T(x,u,v),T(y,u,v),T(z,u,v)}{\mu}_{z,u,v}\end{array}$$(22)
$$\begin{array}{}{\eta}_{T(x,y,z),f(u),f(v)}{\kappa}_{x,y,z}+{\kappa}_{T(x,y,z),f(u),f(v)}={\eta}_{T(x,u,v),T(y,u,v),T(z,u,v)}{\kappa}_{x,u,v}\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}+{\tau}_{T(x,u,v),T(y,u,v),T(z,u,v)}{\kappa}_{y,u,v}+{\mu}_{T(x,u,v),T(y,u,v),T(z,u,v)}{\kappa}_{z,u,v}\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}+{\kappa}_{T(x,u,v),T(y,u,v),T(z,u,v)}\end{array}$$(23)

#### Definition 5.8

*When κ further satisfies κ*_{x,x,x} = 0 *in (23) for any x* ∈ *X*, *we call it a generalized ternary f*-*quandle* 2-*cocycle*.

Recall that the *quandle algebra* of an *f*-quandle (*X*, ⊲, *f*) is a ℤ-algebra ℤ(*X*) presented by generators as in [11] with relations (8), (9), (10), (11), (12), (13), (14).

#### Example 5.9

*Let* (*X*,*T*, *F*) *be a ternary f*-*quandle and A be an abelian group*. *Set η*_{x, y, z} = *τ*_{x, y, z}, *μ*_{x, y, z} = 0, *κ*_{x, y, z} = *ϕ*(*x*, *y*, *z*). *Then ϕ is a* 2-*cocycle*. *That is*,
$$\begin{array}{}\varphi (x,y,z)+\varphi (T(x,y,z),f(u),f(v))=\varphi (x,u,v)+\varphi (y,u,v)+\varphi (T(x,u,v),T(y,u,v),T(z,u,v)).\end{array}$$

#### Example 5.10

*Let Γ* = ℤ[*P*,*Q*,*R*] *denote the ring of Laurent polynomials*. *Then any Γ*-*module M is a* ℤ(*X*)-*module for any ternary f*-*quandle* (*X*, *T*, *F*) *by η*_{x, y, z}(*a*) = *Pa*,*τ*_{x, y,z}(*b*) = *Qb and μ*_{x, y, z}(*c*) = *Rc for any* *x*, *y*, *z* ∈ *X*.

#### Definition 5.11

*A set G equipped with a ternary operator T*: *G* × *G* × *G* → *G is said to be a ternary group* (*G*, *T*) *if it satisfies the following condition*:

*T*(*T*(*x*, *y*, *z*), *u*, *v*) = *T*(*x*, *T*(*y*, *z*, *u*), *v*) = *T*(*x*, *y*, *T*(*z*, *u*, *v*)) *(associativity)*,

*T*(*e*, *x*, *e*) = *T*(*x*, *e*, *e*) = *T*(*e*, *e*, *x*) = *x (existence of identity element)*,

*T*(*x*, *y*, *y*) = *T*(*y*, *x*, *y*) = *T*(*y*, *y*, *x*) = *e (existence of inverse element)*.

#### Example 5.12

*Here we provide an example of a ternary f*-*quandle module and explicit formula of the ternary f*-*quandle* 2-*cocycle obtained from a group* 2-*cocycle*. *Let G be a group and let* 0 → *A* → *E* → *G* → 1 *be a short exact sequence of groups where E* = *A* ⋊_{θ} *G by a group* 2-*cocycle θ and A is an Abelian group*.

*The multiplication rule in E is given by* (*a*, *x*) ⋅ (*b*, *y*) ⋅ (*c*, *z*) = (*a* + *x* ⋅ *b* + *y* ⋅ *c* + *θ*(*x*, *y*, *z*), *T*(*x*, *y*, *z*)), *where x* ⋅ *b means the action of A on G*. *Recall that the group 3*-*cocycle condition is*
$$\begin{array}{}\theta (x,y,z)+x\theta (y,z,u)+\theta (x,yz,u)=\theta (xy,z,u)+\theta (x,y,zu).\end{array}$$

*Now*, *let X* = *G be a ternary f*-*quandle with the operation T*(*x*, *y*, *z*) = *f*(*xy*^{−1}*z*) *and let g*: *A* → *A be a map on A so that we have a map F*: *E* → *E given by F*(*a*,*x*) = (*g*(*a*), *f*(*x*)). *Therefore the group E becomes a ternary f*-*quandle with the operation*
$$\begin{array}{}T((a,x),(b,y),(c,z))=F((a,x)\cdot (b,y{)}^{-1}\cdot (c,z)).\end{array}$$

*Explicit computations give that η*_{x, y, z}(*a*) = *g*(*a*), *τ*_{x, y, z}(*b*) = −2*xy*^{−1}*g*(*b*), *μ*_{x, y, z} = *y*^{−1}*g*(*c*) *and κ*_{x, y, z} = *g*[*θ*(*xy*, *y*, *y*^{−1}) − *θ*(*xy*^{−1}, *y*, *y*) + *θ*(*xy*, *y*^{−1}, *y*) + 2*θ*(*x*, *y*, *e*) − *θ*(*x*, *y*^{2}, *y*^{−1}) − *θ*(*x*, *y*^{−1}, *y*^{2}) + *θ*(*x*, *y*^{−1}, *y*) − *θ*(*x*, *y*,*y*) − *θ*(*x*, *y*, *y*^{−1}) + *θ*(*x*, *y*^{−1}, *z*)].

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